Problem: Solve for $x$ : $ 7|x - 4| - 3 = 6|x - 4| + 7 $
Subtract $ {6|x - 4|} $ from both sides: $ \begin{eqnarray} 7|x - 4| - 3 &=& 6|x - 4| + 7 \\ \\ { - 6|x - 4|} && { - 6|x - 4|} \\ \\ 1|x - 4| - 3 &=& 7 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 1|x - 4| - 3 &=& 7 \\ \\ { + 3} &=& { + 3} \\ \\ 1|x - 4| &=& 10 \end{eqnarray} $ Simplify: $ |x - 4| = 10$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 4 = -10 $ or $ x - 4 = 10 $ Solve for the solution where $x - 4$ is negative: $ x - 4 = -10 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& -10 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& -10 + 4 \end{eqnarray} $ $ x = -6 $ Then calculate the solution where $x - 4$ is positive: $ x - 4 = 10 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& 10 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& 10 + 4 \end{eqnarray} $ $ x = 14 $ Thus, the correct answer is $x = -6 $ or $x = 14 $.